Note that the two conditions on \(G\)  mean that the map \(\phi \)  is isometric - in fact property (A) for \(G\)  gives more. Recall (an equivalent form) of property (A) is operator norm localisation [1]}, which is a sufficient condition to lift operator norm estimates through local liftings (something abstracted from Higson's ideas [2]} and implemented also in Willett–Yu [3]}.
